Discussion Papers No. 173 • Statistics Norway, April 1996

John K. Dagsvik

Aggregation in Matching Markets

Abstract:This paper develops aggregate relations for a matching market of heterogeneous suppliers anddemanders. The point of departure is the analysis of two-sided matching found in Roth and Sotomayor(1990). Under particular assumptions about the distribution of preferences, the present paper derivesasymptotic aggregate relations for the number of realized matches of different types in the presence offlexible contracts (such as a price). Simulation experiments demonstrate that the model also providesexcellent predictions in small populations.

Keywords: two-sided matching models, discrete choice, market equilibrium, marriage models, theGolden Section.

JEL classification: C78, J41

Acknowledgement I have benefitted from comments and criticism on earlier versions of this paper byS. Strom, R. Aaberge, J.J. Heckman, W. Brock, M. Simpson, C. Bollinger, J. Rust, 0. Bjerkholt andparticipants in workshops at the University of Wisconsin, University of Chicago and the NorwegianSchool of Economics and Business Administration. Thanks to Anne Skoglund for word processing andproof reading.

Address: John K. Dagsvik, Statistics Norway, Research Department,P.O.Box 8131 Dep., N-0033 Oslo, Norway. E-mail: jda@ssb.no

1. IntroductionMany important areas of individual behavior involve the search for a partner in a matching

market. Typical examples of matching phenomena are the process of marriage formation, the

admission process of students into colleges, and the matching of employees and workers in the labor

market. A particular important challenge is to obtain a tractable econometric framework for analyzing

matching behavior in a population of heterogeneous agents with preferences that are unknown to the

analyst. This is the topic which will be discussed in this paper.

A game-theoretic analysis of the matching problem relevant for matching markets, started with

Gale and Shapley (1962) and Shapley and Shubik (1972). See Roth and Sotomayor (1990), for an

overview of the literature as well as a theoretical analysis of marriage markets under particular

assumptions about the rules of the game. Becker (1981) applies a matching model to study marriage

and household economics. His concern is to analyze which men are married to which women under

the assumption that the couple derives utility from attributes of the man and the woman.

None of the authors mentioned above consider the problem of developing a mathematically

tractable expression for the probability distribution of the number of realized matches as a function of

parameters that identify the distribution of agents preferences and the size of the relevant population

groups. In the demographic literature, however, several authors have formulated more or less ad hoc

models for the number of marriages formed as a function of the number of unmarried males and

females in each age group (cf. Hoem, 1969, McFarland, 1972, Pollard, 1977, and Schoen, 1977,

Pollak, 1990, Chung, 1994). The only contributions we know of that have attempted to derive a

structural aggregate relationship for the distribution of the number of realized matches, is Tinbergen

(1956).

Our point of departure is a market with suppliers and demanders. Each agent wishes to form a

match with a potential partner which may include specific terms of a contract (such as price, for

example). The menu of contract possibilities is finite and given exogenously. The agents are

heterogeneous with respect to characteristics (attributes). An agent's characteristics affect his own

preferences and enter as attributes in the utility functions of other agents. However, only some of the

attributes are observed by the analyst. Each agent has preferences over all potential partners and over

the contract menu.

In this paper we demonstrate that particular aggregation results are consistent with the

behavioral rules analyzed in Crawford and Knoer (1981). In this case there are no search costs and the

number of agents have full information about the attributes of all potential partners, but have no

information about their preferences. Under particular assumptions about the distribution of the

preferences we show that analytically tractable asymptotic formulae for the number of realized

matches between suppliers and demanders for each combination of observable characteristics follow.

In Dagsvik (1993) a similar setting was analyzed. However, that paper did not provide a behavioral

3

story consistent with the aggregation relations. Note, however, that the aggregate relations derived

below are consistent with any algorithm that produces stable matchings as long as there are no search

costs and the agents have full information about the attributes of all potential partners.

Other approaches than the present one are found in what is called the job search literature.

(See for example Mortensen (1982), (1988), and Diamond and Maskin (1979, 1982), Burdett and

Mortensen (1988), and Burdett and Vishwanath (1988).) The essential difference between the theories

based on optimal search and the first setting described above is that this setting describes behavior in a

market with a finite, known set of potential matching partners. There are no search costs and, ex ante,

the agents have no information about their opportunities in the market. They learn about their

opportunities while they (costlessly) make/receive offers. In contrast, theory formulations based on

optimal search allow for search costs and an unknown set of potential partners. Also the present

formulation takes into account the interaction between different types of agent's that result from the

competition in the market.

The organization of the paper is as follows: In Section 2 the model setting is described and

examples are discussed, and in Section 3 and 4 the matching model with a finite number of agents is

analyzed. In Section 5 we extend the model to allow for flexible contracts and a finite number of

observable (to the econometrician) categories of suppliers and demanders. In Section 6 we allow for

specific correlation patterns in the agents utility functions. In Section 7 some examples are considered,

and in the final section the predictions from the aggregate asymptotic expressions are compared with

"exact" results generated by a series of simulation experiments.

2. Demand, supply and realized matches: description of the gameWe consider a market with suppliers and demanders (agents) that wish to form a match with a

partner. The agents are heterogeneous with respect to unobserved characteristics, called attributes, and

they have preferences over attributes of their potential partners. We shall now discuss the agents

behavior in the matching market and a particular market adjustment process towards equilibrium that

is perceived as taking place in several stages.

Let us first introduce some basic terminology that concerns the rules of the game in the first

setting referred to in the introduction above. The following concepts are borrowed from Roth and

Sotomayor (RS) (1990).

A supplier is acceptable to a demander if the supplier prefers to be matched to the demander

rather than staying unmatched. Consider a matching denoted by lu that matches a pair (s,d) who are not

mutually acceptable. Then at least one of the agents would prefer to be single rather than being

matched to the other. Such a matching 1.1 is said to be blocked by the unhappy agent. A person who

does not obtain a match is said to be self-matched. Consider next a matching such that there exist a

4

supplier s and a demander d who are not matched to another, but who prefer each other to their

assignment at ti (given the rules of the game). The pair (s,d) will be said to block the matching II. We

say that a matching ti is stable if it is not blocked by any individual or pair of agents.

Gale and Shapley (cf. RS) have demonstrated that stable matchings exist for every matching

market. Specifically they described an algorithm called the "deferred acceptance" procedure which

they prove produces a stable matching for any set of preferences provided the preferences are strict.

This algorithm goes as follows: First each supplier makes an offer to his favorite demander. Each

demander rejects the offer from any supplier who is unacceptable to him, and each demander who

receives more than one offer from any supplier rejects all but his most preferred among these. Any

supplier whose offer is not rejected at this point is kept "engaged". At any step any supplier who was

rejected at the previous step makes an offer to his next choice i.e., to his most preferred demander

among those who have not rejected him. Each demander receiving offers rejects any from

unacceptable suppliers, and also rejects all but his most preferred among the group of the new offers

and any supplier he may have kept engaged from the previous step. The algorithm stops after any step

in which no supplier is rejected. The matches are now consummated with each supplier being matched

to the demander he is engaged.

The stability argument goes as follows: Suppose that supplier s and demander d are not

matched to each other, but s prefers d to his own partner. Then demander d must be acceptable to

supplier s, and so he must have made an offer to d before making an offer to his current matching

partner. Since s was not engaged to d when the algorithm stopped, s must have been rejected by d in

favor of someone he (d) liked as least as well. Therefore d is matched to a supplier whom d likes at

least as well as supplier s, and so s and d do not block the matching. Since the matching is not blocked

by any individual or any pair, it is stable. Similarly one could apply a rule where the demanders

making -offers to the suppliers. However, this would not necessarily produce a matching that is equal to

the former one.

To apply the deferred acceptance algorithm requires that the agents' preference lists are known

to the analyst. As mentioned above, our concern is, however, to recover the structure of the

preferences from observations on realized matchings and the size and composition of the population of

agents in the market. In the marriage market literature referred to above it is usually assumed that the

agents are fully informed about potentially available partners and that an agent, if rejected, continues

to make or receive offers until all possibilities in the market have been explored.

A few examples:

*) The marriage market. We have already mentioned the need for establishing a marriage — or mating

function in two-sex demographic models. The relevance of this problem is discussed in Pollak

5

(1990). The results developed below yield tractable functional forms and are therefore well suited

for empirical modelling and analysis.

ii) The labor market. Roth and Sotomayor (1990), see also Crawford and Knoer (1981), have focused

on the matching processes that take place in the labor market. If we abstract from search and

transaction costs the market of medical interns as well as other sectors of the labor market can be

modelled within the framework discussed below.

iii) The housing market. The housing market where people buy and sell houses/apartments can also be

viewed as a matching market. A typical feature of the market of used houses is that the set of

durables (houses) in this market is more or less fixed in the short run. The analogy to the marriage

market is immediately realized from the observation that buyers look for sellers to exchange a

house at some price. Thus the purpose of a match is simply to exchange a house with a "potential"

partner. However, this case is not entirely analogous to the conventional marriage market because

many agents operating in this market are both buyers and sellers: A buyer wants to sell his old

house and a seller has often just made a purchase of a new house. One way to handle this goes as

follows: Note first that it is often the case that the purchase of a new house is made before the old

one has been sold. Thus in this stage the agent is uncertain about the price he may obtain in the

second stage when offering his old house for sale in the market. In the first stage we can still think

of this market as a conventional matching game as described above with the modification that an

agent's preference list is evaluated by taking expected utility with respect to future uncertain sale

price of the old house. Furthermore, the attributes of the sellers in this market are identified by the

attributes of the respective houses offered for sale while the attributes of the buyers may be

variables such as collateral and ability to realize the purchase quickly.

For the sake of focusing on the basic idea of the approach we shall in the next section derive a

model for the probability of realizing a match in the simplest case in which all the suppliers are

observationally identical to the analyst, as are also all the demanders.

3. Aggregation in the special case where suppliers and demanders each areobservationally identical to the analyst

Let N be the number of suppliers and M the number of demanders. In the following we shall

employ small superscripts s and d as indices for a particular supplier, s, and demander, d, and

sometimes capital superscripts, S and D, to indicate supply and demand. Let U sd be the utility of

supplier s of a match with demander d. Let Us° be the utility of supplier s of being self-matched.

Similarly let Vds be the utility of demander d of a match with supplier s and let Vg be the utility of

demander d of being self-matched. We assume that

6

Us sd sd° =aE os U =e,

and

Vo =131 1̀ , V ds

where a and 13 are systematic terms (unknown parameters) that are common to all the suppliers and

demanders, led 1, {cos }, {lids and f__ odanders, respectively, while Ti are i.i.d. random tasteshifters (random to

the observer). Moreover, we assume that

(3.1)

(3.2)

(E sd 5,P (T1

y)=P(cso ).p(ig exp(--)-Y

(3.3)

The assumption (3.3) is consistent with the "Independence from Irrelevant Alternatives" (IIA) axiom,

cf. Ben-Akiva and Lerman (1985). If we had chosen an additive formulation in (3.1) and (3.2) then the

c.d.f. in (3.3) would have had to be replaced by exp (—e ) to be consistent with IIA. This is easily

realized by taking the logarithm of (3.1) and (3.2). The choice between the multiplicative and the

additive formulation is only a matter of taste since they are, from a theoretical point of view,

completely equivalent. Since the set of agents who realize a match is independent of the actual

matching procedure (provided the matching is stable) we do not have to worry about the particular

properties of the algorithm — be it the deferred acceptance algorithm or other rules of the game as long

as preferences are strict. Since the random terms in the utility functions are generated by c.d.f. that are

absolutely continuous it follows that preferences are strict with probability one.

In the final stage (when the matches are consummated), let Ds be the set of demanders that

have not rejected supplier s in previous stages of the game when the suppliers are making the offers.

Let Cd consist of all the offers demander d has received when the suppliers are making the offers.

While Cd clearly consists of all the suppliers that are feasible to demander d, all the demanders in DS

are not necessarily feasible to supplier s. However, if there is a demander d in D S that is not feasible to

supplier s he would be ranked below some other demander that is feasible to s, since otherwise s

would have made an offer to d* and d* would have rejected s at a previous stage. Therefore, D s is

equivalent to the choice set of supplier s in the sense that his choice from DS would be the same as the

choice from his choice set.

Before we plunge into the formal analysis it may be instructive to discuss the intuition behind

the approach. To this end we will assume for a moment that when the population is large one can

ignore variations in the number of consummated matches as a result of variations in the random taste-

shifters. In the formal analysis below we shall prove that this is so. Let m be the number (mean

number) of demanders in Ds and n the number of suppliers in Cd. Since the population in large m and n

will not depend on s and d, respectivley. It follows from (3.1) to (3.3) that the probability that supplier

s shall rank a particular demander in D s on top equals

1 a + m •

Moreover, since there are N suppliers and the suppliers have identically distributed preferences, the

probability that a supplier shall be interested in a match with demander d equals n/N. In equilibrium

we thus have

n 1= •N cc+ m

By symmetry, we also have

m 1=M r3 + n

The probability that a supplier and a demander shall realize a match with each other is the

product of the probabilities that the supplier and the demander are interested in each other. Since the

probability that supplier s shall make an offer to demander d equals n/N and the probability that

demander d shall accept this offer equals m/M the (equilibrium) number of realized matches therefore

equals MN . (m I MXn I N)= mn. Eq. (3.4) and (3.5) determine m and n uniquely as a function of a,

p, M and N.

The treatment above ignores the fact that the sets C d and DS are stochastic and consequently

there are complicated stochastic dependencies between the different opportunity sets {cd, Ds 1. Let us

now therefore turn to a more rigorous argument.

Let I sd =1 if d E D s and zero otherwise, and define I ds =1 if s E C c' and zero otherwise. Then

ms ---= Isk and 11 ddk

are the number of demanders in D s and suppliers in C d. Furthermore, define

m sd sk and n ds dk

kid k*s

which are the number of demanders in DS \ {d} and suppliers in Cd {s}.

Define conditional supply and demand probabilities as

g s =P (U s ' = ax(maxUsr,Uso )IdED s )=E(I ds Id Ds) (3.6)reDs

(3.4)

3.5)

8

D E-P(V ds =-max(Ma3CV dr ,Vod )1SEreC

= E (Isd I s (3.7)

Since the distribution of the utilities does not depend on characteristics of the individual agents, (3.6)

and (3.7) are independent of s and d. Obviously, g s (gD ) is the probability that supplier s (demander

d), in the final stage, shall make an offer to demander d (supplier s) given that demander d (supplier s)

is available in the final stage. From assumptions (3.1) to (3.3) it follows that

1 P (ty sd = max (max u sr syy oU IM sd ,dEp s )==relY a+1+111s

By (3.8) we get

gs = E ( 1 a+1+msd

where the expectation is taken with respect to m sd. Similarly, we get

P (V ds = max(max V dr , Vod )1 SEC d ,reCd

1 +1+n ds

from which follows that

1 Dg1 +(3+ n om

Furthermore we have that

maEm s =Mg D (3.12)

na- E d =Ng s . (3.13)

We shall now study the (aggregate) stable (equilibrium) solution when N and M are large.

Specifically, we shall allow N and M to increase such that N/M tends towards a constant. To obtain

asymptotic results we need the following Lemma.

9

11+T 1 +( 11 – 1)g D

-FM1

<1

E sd1M-1 (3.14)

Lemma 1

Let ti and T2 be any non-negative real number. Then

and

11 < E

1+T 2 +(N– 1)g - 1+T2+n ds1N –1

(3.15)

+ 1-Foc-F(M–N D

The proof in this Lemma is given in the appendix.

Theorem 1

Suppose that N and M increase such that Nhil tends towards a constant, Av 2. Moreover, assume

that a and 13 depend on N such that Fc=-IlimN,a(N) /VII and 13 p(A) /,r-N exist. If the

agents preferences are given by (3.1) and (3.2), then

sdM Ins

pum—r— = plam = u,Ai-400 qM fri—*co M

n ds rldpain— = plim —= v,N—>e* Nri—V N-0. VTI

where

v plim g s IR and u plim g DN-400 M-400

exist and are uniquely determined by the system of equations

v =CT u (3.16)

and

(3.17)

10

P (Vod < max VasSECd

q s =m s

a + ms

qD

n:n

The proof of Theorem 1 is given in the appendix.

Consider now the probability, qS (q D ) that a supplier (demander) shall obtain a match with

any demander (supplier). Clearly supplier s will obtain a match if the utility of being matched to some

demander in Ds is greater than being self-matched. From the assumptions (3.1)-(3.3) it follows that

and

111P(U so <MaXU sd M s) dED a+Ms

(3.18)

Thus

Corollary 1

Under the assumptions of Theorem 1 the (asymptotic) number of realized matches, X, satisfies

the equation

ai3X=(N— XXM— X

(3.22)

of which the only acceptable solution is given by

X=--(af3+M+N—Ika(3+M+N —4MN). (3.23)2

Proof:

From Theorem 1, (3.20) and (3.21) it follows that

N-->co N-4.0 a / ds/M a +ulim q = lim E r--- (3.24)ms/ 4AT

11

and

lim (I D = lim E nd

m--*- (0/ VITT +n d / VR. + v(3.25)

where Ti = lim N, oc/V-M- and 0 = lim N,00 13/jr.

By combining (3.24), (3.16) and (3.25) and (3.17) we obtain

= (1 — liM CI S )11frOTNcie

and

(3.26)

u= q D VAVT3. (3.27)

When (3.26) and (3.27) are inserted into (3.26) we get

(1- lim q D )(1- lim q s )= v TEI3- lim q s . (3.28)N-400

Since urn q s =11, 2 lim q D and X =N q s , (3.22) follows. Since (3.23) is the only solution of (3.22)N-÷00 N4-4.0

within (0, min (M,N)), it is the only acceptable solution.

Q.E.D.

From (3.23) we realize that when N=M and a and 13 are close to zero then X is close to N

when N is large. At first glanse this may seem surprising, since the population of suppliers and

demanders have the same size. The explanation is that since the utility functions have i.i.d. random

tasteshifters then for sufficiently large N the probability that a supplier will find a very attractive

demander which ranks the supplier on top (among all suppliers) will be close to one.

4. A special case: The Golden SectionIt has long been realized that certain shapes of rectangle seem to the human eye to be

aesthetically more satisfactory than others. Indeed, given a large range of rectangular shapes to choose

from, most people, it is said, will tend to choose as most satisfactory one which length bears to its

width the same ratio as the sum of the length and the width bear to the length alone. The resulting ratio

is called the Golden Section ((p) and it is determined by the equation

cp 2 = +1 (4.1)

12

i.e., 9=04-19/2.

The Golden Section is exhibited in the Athenian Parthenon and a number of other buildings of

classical antiquity and it is also found in the Egyptian Great Pyramid as the ratio between the slope-

height and the half-base (within .001 of the Golden Section). Moreover, it is found at the entrance of

the tomb of Ramses IX and on the walls of the colonnade of Amon in the Temple of Luxor (cf.

Schwaller de Lubicz, 1985 and Lemesurier, 1977).

The Golden Section is also linked to the so-called Fibonacci Series where each number equals

the sum of its two predecessors. It is found with surprising frequency in nature, for example in pattern

of plant growth, in flower-petal arrangements, in the laws of Mendelian heredity and in the ratios

between planetary orbits.

It is intriguing that the (inverse) Golden Section also emerges as a solution of (3.15) and (3.16)

in the following special case with M=N and "Cc = D=1, which means that a=i3=ViSf . Recall that the

assumption that a/4M-- and 13/4R tend towards constants, -a- and TS , when M and N increases mean

that the fraction of people that prefer to be self-matched asymptotically remains the same when the

population grows. When N=1 the last assumption, -es,c- = 3 =1, means that the probability of preferring a

match over being self-matched is equal to 1/2. Thus, when only one potential partner is present the

agents are, on average, indifferent between the two alternatives "being matched" and "self-matched".•

Under these assumptions it follows that g s =gD and q s =c1D. Form (3.15) and (3.16) we get that

u = v =1/(p. From (3.21) we obtain that

(IS =q° =1/92. (4.2)

The probability of being self-matched equals

1- q s =1 -1/9 2 =1/9. (4.3)

The last equality in (4.3) follows from (4.1).

5. The general case with flexible contracts and several observable categories ofsuppliers and demanders

In this section we shall modify the description in Sections 2 and 3 so as to allow for flexible

contracts. Relevant examples are tuition fees and grades in the market for education, and wages and

non-pecuniary conditions in the labor market.

One particular rule of the game we shall consider in the present section is a simple extension

of the deferred acceptance algorithm discussed in Section 3: Let us modify the list of rank orderings by

13

considering each supplier's (demander's) list of rank ordering of all possible combinations of contract —

and demander (supplier) attributes. Thus, when a supplier (say) makes an offer to a demander this

offer includes specific contract terms. With this modification the game proceeds as the deferred

acceptance algorithm described in Section 2. The algorithm described above has been analyzed

theoretically by Crawford and Knoer (1981).

In the present section we also assume that a subset of the attributes are observable to the

econometrician such that the agents can be grouped into a finite number of (observable) categories.

Specifically, let NI; be the number of demanders of type j, j=1,2,...,D, and N ; the number of suppliers

of type i, i=1,2,...,S. The total number of possible contracts is also finite and equal to W, the total

number of suppliers and demanders are N and M. The assumption of finite W is made for simplicity

and can easily be relaxed.

Let Ulf (w) be the utility of supplier s of type i of a match with demander d of type j with

contract w. Let Usio be the utility of supplier s of type i of being self-matched. We assume that

U sio =aio E sio , U (w) =

a;1 (w)eld (w) (5.1)

sdwhere fa (w)} and { } are systematic terms and 1E ij (w)} as well as {c;° } are i.i.d. random taste-

shifters.

On the demand side the description is completely analogous. Thus

Vido = b oo o,Vc!: (w) = b ii (w)Tilis(w) (5.2)

is the utility function of demander d that corresponds to (5.1).

Let D (w) be the equilibrium set of demanders of type j that has not rejected a match offered

by supplier s under contract w. Similarly, let C id; (w) be the equilibrium set of all the offers of type i

demander d of type j under contract w has received. If the rules of the game is given by the deferred

acceptance algorithm then {C idi (w)} and IN (w)} correspond to the choice sets in the final stage of

the game. However, as discussed in Section 2 a more general interpretation is possible. Let m (w)

and n idi (w) be the number of demanders in Dsi• (w) and suppliers in C d(w) . We shall investigate

below the conditions under which (aggregate) market equilibrium exists when the population of

suppliers and demanders are large.

Similarly to (3.3) we assume that eijd (w), eidis (w), ciao, i=1,2,•••,

d=1,2,...,M 3 , j=1,2,...,D, w=1,2,...,W, are i.i.d. with

14

p ( .)1).= PO: ..y)=P(Elcs 5..y)=PRI0 exp (5.3)

Consider the behavior of supplier s conditional on the choice set D; (w) . Let (w) be the

probability that supplier s of type i will prefer demander d of type j under contract w. We shall call

fg; )} the conditional supply probabilities. Specifically, for d E D; (w)

qigjd (w)=max(max( max Urki(r)),U10 ) cleDl(k,r (r)

)Ja (w)

I d (W)

a ic, +a ki (w)-qm sii (w)-1)a jj (w)+ ml(r)ai ( )(k,r)*(j,w)

(

=E cIEN(w)a ik (r)ml( r)+a io

aii(w)

k r>0

The derivation of (5.4) is completely analogous to the derivation of the choice probabilities of

the extreme value random utility model, see Ben-Akiva and Lerman op cit. Similarly, the conditional

demand probabilities are given by

(

gij (W

(

=E (5.4)

gP(w)=E(w)

b io nik(r)bik r>0

s€C1(w)j

•

(5.5)

To facilitate further calculations it will be convenient to introduce additional notation. Let

(w) =1 if d E (w) and zero otherwise and define m i• (w)= E mlj (w) and n ii (w) =End (w) .

Obviously we have

m ij (w)EEml(w)=M i ei (w) (5.6)

n ii (w) En"; (w) = N i g s ( ).ij (5.7)

The equations (5.4) to (5.7), hold under the deferred acceptance game as well as in other games that

produce stable matchings. By letting 11•T i and 11■4 j increase towards infinity one can examine the

asymptotic properties of this system, similarly to the analysis in Section 3.

15

a ij (w)

S aij (w)gij (w)=

a ik (r) E (Inlk d E (w)k r>0

+a io

Obviously, E I d (w) = m ij (w)/M i , since

NI ;

rr,i (w)= I (w). (5.8)u=1

We have

Also, for r# w,

E m (w) d E (W)) = E m (w) I (w) =1)

07(w)gid (w)=1)=1+ EI7(w)=1+ M i —1u*d

ril- (W)

M j

(5.9)

E mlj (r) dEN(w))=11 E (r) (w)=1 = m i - 1 M ijr

M j(5.10)

because

E (w)=1 =

due to the fact that Id (k) can only be different from zero for one k. Finally, we have

E (m sik (w)) = m ik (r) (5.11)

when k # j. Now by a first order Taylor approximation and taking account of (5.9), (5.10) and (5.11),

it follows from (5.4) that

(5.12)

a ik (r)m ik (r)+a ij (w)—I a ki (r) ii (r)/M i +a io

k r>0 r>0

Similarly, it follows from (5.6) that

g jDi b ii (w) (5.13)

k r>0

(r)n ik (r)+b ii (w)-11 ii (r)n ji (r)/N i +r>0

Since {N ; }and {M i } are assumed to be large, it follows that

(m ij (w) + 1)/m (w) (w) + 1)/n j; (w) Es 1, and (N i —1)/N 1 — , and that the

16

approximations in (5.12) and (5.13) are close. We shall therefore henceforth replace by "=". From

(5.12), (5.13), (5.6) and (5.7) we therefore get

M i b ii (w)

B i(5.14)

n ji (w (5.15)

for i=1,2,...,S, j=1,2,...,D, and w=1,2,...,W, where

A i =a ic, + (r)mik(r) (5.16)k r>0

b.oB j. = +I b jk (r)n i (r).Jk r>0

(5.17)

We shall discuss the problem of existence and uniqueness of a solution of (5.14) to (5.17) in

the next section. Another crucial problem is how the approximation error in (5.12) and (5.13) depend

on 1N i and {M i }. We have conducted a series of simulation experiments to throw light on this

issue. The results (cf. Section 9) show that, on average, the approximation is rather small.

Let us now consider the probability of realizing a match with a particular contract. Given that

supplier s belongs to type i and demander d belongs to type j, the probability that a match between a

supplier s of type i and a demander d of type j under contract w shall occur is equal to the product

between the respective conditional supply and demand probabilities, g sij (w) and g iDi (w). The

probability that supplier s of type i shall obtain a match under contract w with any demander of type j,

s -w, iq ij ( ), is therefore equal to

k = (w)ei (A') m ; • (5.18)

Similarly, the mean number of matches under contract w where the supplier is of type i and the

demander is of type j, X ii(w), equals

X.. (w = g (w) g)j?i N. M • (5.19)

Consequently, we obtain

X Ii

a ii (w)b ji (w)M i N i

A. B j(5.20)

17

for i=1,2,...,S, j=1,2,...,D , w=1,2,...,W.

When (5.14) is inserted into (5.16) and (5.15) is inserted into (5.17) we obtain that {A i and

{13 .i I satisfy the equations

Ai =a io +Cik Mk

B k(5.21)

B• =b• +j joC kj N k

A k(5.22)

where

= a ii (r)b ii (r)•r>0

The respective mean number of self-matched suppliers and demanders, X sio , X iDo , are given by

Xs ai0 N i

A i

and

b • MX D = jo Bi

From Theorem 2 below it follows that the equations (5.21) and (5.22) have a unique solution.

(5.23)

(5.24)

(5.25)

6. ExtensionsIn the derivations above we assumed that the random components of the utility functions were

independent and extreme value distributed. We shall now relax some of these assumptions and we start

with the simple setting discussed in Section 3.

6.1. Observationally identical suppliers and demanders

We shall now allow the taste-shifters associated with potential partners to be correlated. This

may be desirable because there may be unobservable factors that affect the utility for potential partners

which are correlated (across the potential partners). To this end we now postulate that the vector

(esi,Es2,...,csm) is distributed according to a type I multivariate extreme value distribution given by

18

P(Esd < = exp y (6.1)

(cf. Ben-Akiva and Lerman, 1985), where 0 1 e 0,1] is a constant. The particular version given in (6.1)

takes into account that the players are "anynomous" to that the taste-shifters are exchangable.

Moreover, 0 1 can be interpreted as

con, sd Esk = stsq (6.2)

for k # d. Similarly, we assume that

P ods <Ys)) = exp — (6.3)

where 0 2 E (0,1]. However, we assume that E so is independent of esd and ,nds is independent of ig ,

where the c.d.f. of e; andng is given by (3.3). Thus when e i and 02 are less than one it means that

different potential partners are perceived as more "similar" to the individual than the alternatives

"being selfmatched" and "being matched" to some potential partners. In case O i and 02 are close to

zero the potential partners appear to be almost identical to the individual. Under assumptions (3.1),

(3.2), (6.1) and (6.3), it follows readily that

P (Ird = max (max1r , U sorelY

sa ,d(1+ms ler-1

D 5 )= a+(l+rnsd )

(6.4)

P (V ds = max (max V dr , VodreC

(l+n )O2-1ch ,s C d )

±(1 +n )(6.5)

Similarly to (3.9) and (3.11) we therefore obtain

gs = E

f(1+ msd

a + (1, ± mut \(6.6)

gp = E (11-ndsr2-1 \[

13+(l+nds)(6.7)

19

M — Xn = (

13X ) 1/e2 (6.12)

N—Xl e2 ie M—X 1'1 le _Xa

(6.13)

a+mn =Nm er-1 E( 1 (6.14)

where the expectation is taken with respect to m sd and nth . As in Theorem 1, it can be demonstrated

that the asymptotic values of m and n are given by

+ m sd e, -1nal•Ig s =NE

N m e i -1

a+0A-msdr a+mel

and

(6.8)

I + n dl e2—

R ± h ±n dsrY"' k

m.:-,..MgD =MEMn e2-1

13+n e2(6.9)

Asymptotically, the probability that supplier s shall obtain a match with any demander equals g sm,

because m is (asymptotically) the number of demander available to supplier s. Since there are N

suppliers the total (asymptotic) number of matches equals

X=Ng s m=nm. (6.10)

From (6.8) and (6.9) we therefore obtain that

nd "X — X)

(6.11)

Let 0.03 1 +0 2 —0 1 0 2 . Consequently, (6.10), (6.11) and (6.12) imply that X must satisfy the equation

which generalizes (3.22). It is easily verified that eq. (6.13) has only one positive solution.

Let us next discuss a further modification of the distributional assumptions. Suppose now that

a and i3 are positive random variables. Then the analysis above becomes entirely similar and we obtain

(6.6) and (6.7) where the expectation now is taken with respect to both a and Insd in (6.6) and 13 and nds

in (6.7). The resulting asymptotic expressions are analogous to (6.8) and (6.9), and are given as

20

2+n(

m=Mne2-1E 1 (6.15)

where the expectation is taken with respect to a and 13. The motivation for allowing a and 13 to be

random is that the population heterogeneity may be such that the distribution of the preferences cannot

be represented by the extreme value distribution alone. It is easily verified that (6.14) and (6.15) have

a unique solution for n and m, and X=mn is therefore uniquely determined as a function of el, 02 and

the distribution of a and 0.

6.2. The general case

This subsettion extends the framework developed in Section 5 so as to permit correlated taste-

shifters as introduced above. Specifically, we assume that

(qi(w).Yd)) = exp(

Yd -lie, (6.16)

P 0.11:(w)5_ys))= exP[—(li (6.17)

while £ (w) and 2 11E(W) as well as tis (w) and - sa (W) are assumed independent when

(s, a, b) # j) , and (d, j, i) # (a, b, a), respectively. Similarly £ 10 and jdo are independent of the

other taste-shifters as well as independent across agents. The respective c.d.f. are the same as in (5.3).

By a straight forward application of the Generalized Extreme Value model, cf. Ben-Akiva and Lerman

(1985), p. 123, it follows readily that (5.14) to (5.17) extend to

N b ij (w)n ii (w) (32-1m..(w)= (6.18)

M J a-n

0 1 -1

(6.19)

where

21

and

A i = a io + ail, (r) M ik

k r>0

=b io +Ef b. (r)n. (r) e2jk jk •k r>0

(6.20)

(6.21)

When (6.18) and (6.19) are combined we obtain

N 1 b ji (w) M j a ij (w)mid (w)

Ai

and

(6.22)

1/9

n ji (w) =M. a.. w N i b ii (wi

B(6.23)

where 0=0 1 +13 2 - 9 1 0 2 . Furthermore, (6.21) and (6.22) yield

N i b ii (w)X ii (w)= m ii (w)n ii (w)=

B j

)0110 02/e

(6.24)

s = ai° NiA i

and

X D - j°Jo - -B j

which extend the model derived in Section 5.

(6.25)

(6.26)

Theorem 2

Suppose a io > 0 and b ./0 > 0 for all i and j. Then the equations (6.20) to (6.23) determine

{mu (w)} and In J, (w)} uniquely.

The proof of Theorem 2 is given in the appendix.

22

= a io aik(r)mik( )

k r>0

a (w) m (w)(7.1)

7. ComplementsIn this section we collect some complements and remarks that have relevance for potential

applications.

7.1 Note that by (5.16) we can express (5.18) as

Suppose now that the demands {TN (w)} are observable. Then (7.1) suggests a convenient way of

estimating the preference terms {a ki (w)} from microdata on realized choices in a matching market.

Specifically, if vii(w) = logaii(w) is specified as a parametric function in explanatory variables, then the

preference structure can be estimated by a multinominal logit type of analysis in which the structural

terms vii(w) must be adjusted by adding logm ij(w).

7.2 Suppose now that {IN (w)} are not observable but that precise estimates on the number of

demanders that are selfmatched, X 11)0 , are available. With bio=1, (5.25) yields

X P Mo =B ;

!

which by (5.21) and (5.20) imply that

a -1-y a ik (r)b ki (r)X koJo•

k r>0

If a parametric specification of loga ii(w) + logbii(w) is chosen the model can, as above, be estimated by

a multinomial logit type of analysis on microdata in which the structural terms must be adjusted by the

logarithm of the numbers of self-matched demanders of the respective types.

8. Simulation experimentsRecall that the models derived above are only assumed to hold asymptotically, i.e., when the

respective population groups are large. It is therefore of considerable interest to analyze how the

different model versions perform in small population groups. To this end, we report results from a few

a ii (w)b ii (w)X io(w) =ij

(7.2)

(7.3)

23

selected simulation experiments. For more detailed simulation results we refer to Dagsvik and

Johansen (1996). The simulation experiments are carried out as follows: First independent taste-

shifters are drawn from the extreme value distribution (3.3). From these draws and selected values of

the systematic terms of the agents' utility functions, preference lists are assigned to every agent.

Subsequently, the deferred acceptance algorithm is simulated, which produces the matchings. Tables 1

and 2 below display the results. In Table 1 we report the results from simulating the model discussed

in Section 3. We have performed simulations with 14 different sets of population sizes and preference

parameters. For each set of parameters the experiments were replicated 1000 times to obtain precise

estimates of the respective means and standard deviations.

Table 1. Simulation results with observationally suppliers and demanders

Preferences Population size Number of matches

a N M Predicted Simulated Standarddeviation

1 7 7 50 50 19.27 17.73 2.972 7 10 50 150 31.44 30.05 3.293 1 2 60 80 55.48 54.56 2.014 1 1 30 15 14.11 13.58 1.105 4 1 30 20 15.64 14.67 1.746 6 8 15 20 3.79 3.32 1.547 9 3 10 15 3.07 2.57 . 1.278 20 1 15 90 11.94 10.05 1.789 3 4 20 40 13.73 12.72 1.9810 2 1 10 5 3.78 3.32 1.0111 1 5 80 40 35.92 33.78 2.1012 30 7 30 70 6.93 6.25 2.0313 3 2 20 20 11.64 10.57 1.8514 2 5 8 15 4.16 3.57 1.29

The fifth column in Table 1 displays the predicted number of matches that follows from

(3.23). In coloumns 6 and 7 we report the mean of the number of simulated matches (across the 1000

replications) and the standard deviation of the number of matches. The table shows that the model

provides excellent predictions.

In Table 2 we report the results from a more general experiment in which there are two

observed types of suppliers and demanders. We report the results from four experiments where each

type of experiment is replicated 100 times.

24

Table 2. Simulation results with two types of suppliers and demanders

Experiments 1 2 3 4

an 8 20 1 4a12 2 3 0.2 0.5a21 3 8 0.2 3

Preferences a22 2 1 2 2b11 1 5 8 1b12 1 2 1 1b21 2 3 1 1b22 3 1 3 2

N 1 20 30 20 15Population N2 15 10 60 20sizes M1 30 5 10 10

M2 8 20 30 15

Predicted X11 17.16 4.33 9.23 6.40Simulated X11 17.03 4.88 8.51 6.68Predicted X12 2.24 17.85 0.34 2.1.7

Mean number Simulated X12 2.45 16.72 0.49 1.88of matches Predicted X21 9.29 0.66 0.66 3.35

Simulated X21 9.84 0.12 1.25 3.07Predicted X22 4.84 1.89 29.49 12.15Simulated X22 4.26 2.35 29.21 12.34

S11 1.41 0.32 1.65 1.04Standard s12 1.24 1.23 0.83 0.72deviations S21 1.19 0.32 1.43 1.00

S22 1.21 0.97 0.94 0.93

The predictions reported in Table 2 are obtained from (5.20), (5.21) and (5.22) with the

number of feasible contracts equal to one and with a io = b jo =1. Thus the (mean) number of matches

predicted, where the supplier is of type i and the demander is of type j, equals

N i M i a ij b pi

= A i Bi(8.1)

where A i and Bi are determined by

and

for i, j =1,2.

2

A i aik b ki M k /B k

k=1

2

B j = 1 + kj b jk N k /Ak

k=1

(8.2)

(8.3)

25

Similarly to the results reported in Table 1, the results displayed in Table 2 also demonstrate

that there is practically no bias in the theoretical predictions from the model.

9. ConclusionIn this paper we have discussed the problem of aggregation in particular matching markets.

Under specific assumptions about the matching game and the distribution of the agents preferences we

have obtained tractable expressions for key aggregate relations, such as the number of realized

matches. We have discussed the deferred acceptance algorithm as a possible behavioral story that yield

stable matchings. This algorithm has been extended to allow for a finite menu of flexible contracts,

such as a price — or a wage rate. This is of interest in many fields of application, such as the labor

market and the market for education.

Since the aggregate relations are asymptotic ones we have conducted a series of simulation

experiments to assess the degree of bias in small populations. The simulation experiments indicate that

the prediction bias in small populations is negligible.

26

Dks))

(A.6)

• •

Appendix

Proof of Lemma 1

Note first that the moment generating function of rk conditional on Ilks, equals

E(zisk n ks )=1— iD(nks)+ze(nks).

Since Isk, k=1,2,..., are independent when n ks are given we get

qz1= (zig In ks)=E110—"On ksk*d k*d

for Z E [0,1] , where

1 k p ( )= 1+0+n

If X is a non-negative discrete random variable it is immediately veryfied that

)

= .1 z t-1 E(z x )clz(1+T+X

for any positive constant T. Consequently, (A.2) and (A.4) imply that

1+ ,,c + m sd (

1f it krij—iD(nks)-1-zip(nks))dz . A.5)

Observe next that ez -1 — z 0 when z E [0,1]. To realize this we note that e' — z is decreasing in z

and consequently i's minimum value is zero and it is attained at z=1. Therefore

(A.1)

(A.2)

(A.3)

(A.4)

0..... iD(n ks) ±z iD( ksn ))clz exp —

k*d0 0

(z — )1,k*d

1— exp(—T — g l) ksk*d fd (nls

, n 2siD(n

kid

where

fd (n ls n 2s Ms ) = (A.7)1

k*d 1+13±nics

27

Ft+ u< v <

—

f3+v

(A.11)

and

From (3.6) and (3.7) it follows that

E(msd = E (I* I s EA k )= —1)gD (A.8)Ic*cl

and

E(n ds)= E(I dk IcleB k HN—Og s . (A.9)Ic*cl

Since the function f is concave in (n is n, 2s ,...) on R m.:4 , it follows from Jensen's inequality, (A.5)

and (A.9) that

f is 2s Ms

< fd En's, En 2s ,..., En Nis =1 (A.10)

M —1 t+

1+13+(N —1)g s

which proves the right hand side of (3.14). The left hand side of (3.14) follows directly by the

application of Jensen's inequality and inserting for E(ms d ) given by (A.8). The proof of (3.15) is

completely analogous.

Q.E.D.

Proof of Theorem 1:

Put i 1 = a and it 2 =13 in (3.14) and (3.15). When N and M are large Lemma 1 implies that for

<u<13+v

(A.12)

a+u

By inverting the right hand side of (A.11) we get

— V—cc +=—. --v [3+v

and similarly from (A.12) we get

(A.13)

28

(A.14)

By multiplying (A.13) by v and (A.14) by u we obtian that

13-V-1 V+ -

13+vrxv+v -1 (A.15)

Let

-1 Vyf + —a +u(A.16)

X= + + u

— — (A.17)

and

From (A.12) and (A.11) we have that

0 —y=v+ av—► .13+ v

(A.18)

I T3u<b-cv and — —0±vFc+u

which imply that

x Ar l + rxv — — N' (A.19)

and

y5v+13u—ircv—v -1 . (A.20)

But (A.19) and (A.20) imply that x+y 5,0. Since by (A.15) and (A.16), x 0 and y 0 , we conclude

that x = y = 0. Consequently, the inequalities on the right hand sides of (A.11) and (A.12) reduce to

equalities, i.e.,

v= (A.21)

29

u= — vFt+u

(A.22)

When (3.16) is inserted into (3.17) and (3.17) is inserted into (3.16), respectively, we obtain (A.22)

and (A.21). It is straight forward to show that (A.21) and (A.22) have only one positive solution.

It now only remains to prove that msc l i,fivi and n ds i■I converge in probability toward u

and v, respectively. From (3.14), (3.16) and (3.17) we get with it = NrCI

1 1 < lim E •t+u 1v1->- t+msd/VM -1

er+ (3+v

But by (3.17) the right hand side of (A.23) equals (ti + u) -1 which by (A.23) implies that

lim E 1 1

M-->°° (t+msd i4R) T+u

(A.23)

(A.24)

for any positive it. The left hand side of (A.24) is the Stieltjes transform of the c.d.f. of ms d/Nrgi (see

Widder, 1941) which has the same uniqueness and convergence properties as the Laplace transform.

Thus since the right hand side of (A.24) is the Stieltjes transform of a degenerate distribution it follows

that

m sd j\TI u .

The proof that

rids/4N

is completely analogous to the proof above.

Q.E.D.

Proof of Theorem 2:

Let F = (F', F") , where F' and F" are mappings F', F" R2sDw R2SDW The components

of F' are given by

30

1— —

e logl b +1„ bik(r)exp (02 Y jk

k r>0 ))J (A,25)

a (x, y) 02 b j (r) exp

a Y.* (r) 013j

Y jk (0)(A.30)

Let

Fi'j,, (x, y) = log [(N i b ji (w)) 110 (M j a ij (w)) (02-1)10 1

— 02 ) log (a io + a ik (r) exp x ik (r))k r>0

and the components of F" are given by

FE W (x, y) = log [(M i a ij (w)) 110 (N i b ji (w)) (011)10 ]

—

01

aio aik(r)expOi x ik (0)k r>0

(A.26)

e

l) log (kJ() + b jk (r) exp (02 y ik (r))

\— e

We realize that (A.25) and (A.26) equal the logarithm of the right hand side of (6.22) and (6.23) with

x ij (w)= log m ij (w) and y ji (w) = log n ji (w). Note that there is no loss of generality in assuming that

x =7: (w)} and y E{y ji (w)} are restricted to a compact set, A (say). To realize this, note that by

(6.20) and (6.21)

a io 5...A i aik(r)Mk (A.27)k r>0

and

boo (r)Nk (A.28)k r>0

which, by (6.22) and (6.23), imply that m ii(w) and nji(w) are bounded from above and bounded away

from zero. Hence, x and y can be restricted to a compact set. We have

k r>0

a N. (x, y) (1-02)81 a ik (r) exp xik (

a X ik (r) 8A.

))

(A.29)

31

< — a i0 ) 0 2 (ift j by)) (e l —

i B

aF;;,„, (x, y)

ax ik (r)

api;w (x, y)a Yik (r)

(A.31)e 2 +02 )x= K <1.

(vwx' (x ' Y) "" K I —y i I+Kmax fikr — Ik, r>0

(A.35)

and

K max a. 1 b.0

).A i

Since ; and t j are bounded and positive, K must be less than one. Hence, (A.29) and (A.30) imply

that

Similarly, it follows that

auw (x, y)

ax ik (r)

aFfiw (x,y)

a jk (r)K <1. (A.32)

Let II-II be the norm defined by 114= max k IX k I, for x E R 2SDW . Now by the mean value theorem for

vector fields we have

') — (x,yr>0

(

a Fi'iw (x * , y

a y ikr ikr — x i A.33)

a Ffi'w (x * ,y * ))+ —Y -kr)

a Y kr j Ffi'w y') — (x, y) =

k r>0 (x ikr x i (A.34)

where (x', y') and (x,y) are two elements in R2SDW and (x * , y * ) is a point on the hyperplan through

(x', y') and (x,y). From (A.31), (A32), (A.33) and (A.34) we get

which imply that

FiT„, y') Fi'i'w (x, y) <<K maxk, r>0

X ikr X i + K maxk, r>0

(A.36)

IIF(x F 'Y )II 11(x' Y)II. (A.37)

32

Consequently, F is a contraction mapping. From Blackwell's Theorem (Blackwell, 1965) we then

know that the equation

(x,y) = F(x,y) (A.38)

has a unique fixed point in A.

Q.E.D.

33

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35

Issued in the series Discussion Papers

42 R. Aaberge, 0. Kravdal and T. Wennemo (1989): Un-observed Heterogeneity in Models of Marriage Dis-solution.

43 K.A. Mork, H.T. Mysen and 0. Olsen (1989): BusinessCycles and Oil Price fluctuations: Some evidence for sixOECD countries.

44 B. Bye, T. Bye and L. Lorentsen (1989): SIMEN. Studiesof Industry, Environment and Energy towards 2000.

45 0. Bjerkholt, E. Gjelsvik and 0. Olsen (1989): GasTrade and Demand in Northwest Europe: Regulation,Bargaining and Competition.

46 L.S. Starnbol and K.O. SOrensen (1989): MigrationAnalysis and Regional Population Projections.

47 V. Christiansen (1990): A Note on the Short Run VersusLong Run Welfare Gain from a Tax Reform.

48 S. Glomsrod, H. Vennemo and T. Johnsen (1990): Sta-bilization of Emissions of CO 2: A Computable GeneralEquilibrium Assessment.

49 J. Aasness (1990): Properties of Demand Functions forLinear Consumption Aggregates.

50 J.G. de Leon (1990): Empirical EDA Models to Fit andProject Time Series of Age-Specific Mortality Rates.

51 J.G. de Leon (1990): Recent Developments in ParityProgression Intensities in Norway. An Analysis Based onPopulation Register Data

52 R. Aaberge and T. Wennemo (1990): Non-StationaryInflow and Duration of Unemployment

53 R. Aaberge, J.K. Dagsvik and S. Strom (1990): LaborSupply, Income Distribution and Excess Burden ofPersonal Income Taxation in Sweden

54 R. Aaberge, J.K. Dagsvik and S. Strom (1990): LaborSupply, Income Distribution and Excess Burden ofPersonal Income Taxation in Norway

55 H. Vennemo (1990): Optimal Taxation in Applied Ge-neral Equilibrium Models Adopting the ArmingtonAssumption

56 N.M. Stolen (1990): Is there a NAIRU in Norway?

57 A. Cappelen (1991): Macroeconomic Modelling: TheNorwegian Experience

58 J.K. Dagsvik and R. Aaberge (1991): HouseholdProduction, Consumption and Time Allocation in Peru

59 R. Aaberge and J.K. Dagsvik (1991): Inequality inDistribution of Hours of Work and Consumption in Peru

60 T.J. Klette (1991): On the Importance of R&D andOwnership for Productivity Growth. Evidence fromNorwegian Micro-Data 1976-85

61 K.H. Alfsen (1991): Use of Macroeconomic Models inAnalysis of Environmental Problems in Norway andConsequences for Environmental Statistics

62 H. Vennemo (1991): An Applied General EquilibriumAssessment of the Marginal Cost of Public Funds inNorway

63 H. Vennemo (1991): The Marginal Cost of PublicFunds: A Comment on the Literature

64 A. Brendemoen and H. Vennemo (1991): A climateconvention and the Norwegian economy: A CGE as-sessment

65 K.A. Brekke (1991): Net National Product as a WelfareIndicator

66 E. Bowitz and E. Storm (1991): Will Restrictive DemandPolicy Improve Public Sector Balance?

67 A. Cappelen (1991): MODAG. A Medium TermMacroeconomic Model of the Norwegian Economy

68 B. Bye (1992): Modelling Consumers' Energy Demand

69 K.H. Alfsen, A. Brendemoen and S. Glomsrod (1992):Benefits of Climate Policies: Some Tentative Calcula-tions

70 R. Aaberge, Xiaojie Chen, Jing Li and Xuezeng Li(1992): The Structure of Economic Inequality amongHouseholds Living in Urban Sichuan and Liaoning,1990

71 K.H. Alfsen, K.A. Brekke, F. Brunvoll, H. Luris, K.Nyborg and H.W. S2ebo (1992): Environmental Indi-cators

72 B. Bye and E. Holmoy (1992): Dynamic EquilibriumAdjustments to a Terms of Trade Disturbance

73 0. Aukrust (1992): The Scandinavian Contribution toNational Accounting

74 J. Aasness, E. Eide and T. Skjerpen (1992): A Crimi-nometric Study Using Panel Data and Latent Variables

75 R. Aaberge and Xuezeng Li (1992): The Trend inIncome Inequality in Urban Sichuan and Liaoning, 1986-1990

76 J.K. Dagsvik and S. StrOm (1992): Labor Supply withNon-convex Budget Sets, Hours Restriction and Non-pecuniary Job-attributes

77 J.K. Dagsvik (1992): Intertemporal Discrete Choice,Random Tastes and Functional Form

78 H. Vennemo (1993): Tax Reforms when Utility isComposed of Additive Functions

79 J.K. Dagsvik (1993): Discrete and Continuous Choice,Max-stable Processes and Independence from IrrelevantAttributes

80 J.K. Dagsvik (1993): How Large is the Class of Gen-eralized Extreme Value Random Utility Models?

81 H. Birkelund, E. Gjelsvik, M. Aaserud (1993): Carbon/energy Taxes and the Energy Market in WesternEurope

82 E. Bowitz (1993): Unemployment and the Growth in theNumber of Recipients of Disability Benefits in Norway

83 L. Andreassen (1993): Theoretical and EconometricModeling of Disequilibrium

84 K.A. Brekke (1993): Do Cost-Benefit Analyses favourEnvironmentalists?

85 L. Andreassen (1993): Demographic Forecasting with aDynamic Stochastic Microsimulation Model

86 G.B. Asheim and K.A. Brekke (1993): Sustainabilitywhen Resource Management has Stochastic Conse-quences

87 0. Bjerkholt and Yu Zhu (1993): Living Conditions ofUrban Chinese Households around 1990

88 R. Aaberge (1993): Theoretical Foundations of LorenzCurve Orderings

89 J. Aasness, E. Biorn and T. Skjerpen (1993): EngelFunctions, Panel Data, and Latent Variables - withDetailed Results

36

90 I. Svendsen (1993): Testing the Rational ExpectationsHypothesis Using Norwegian Microeconomic DataTesting the REH. Using Norwegian MicroeconomicData

91 E. Bowitz, A. Rodseth and E. Storm (1993): FiscalExpansion, the Budget Deficit and the Economy: Nor-way 1988-91

92 R. Aaberge, U. Colombino and S. Strom (1993): LaborSupply in Italy

93 T.J. Klette (1993): Is Price Equal to Marginal Costs? AnIntegrated Study of Price-Cost Margins and ScaleEconomies among Norwegian Manufacturing Estab-lishments 1975-90

94 J.K. Dagsvik (1993): Choice Probabilities and Equili-brium Conditions in a Matching Market with FlexibleContracts

95 T. Komstad (1993): Empirical Approaches for Ana-lysing Consumption and Labour Supply in a Life CyclePerspective

96 T. Komstad (1993): An Empirical Life Cycle Model ofSavings, Labour Supply and Consumption withoutIntertemporal Separability

97 S. Kvemdokk (1993): Coalitions and Side Payments inInternational CO2 Treaties

98 T. Eika (1993): Wage Equations in Macro Models.Phillips Curve versus Error Correction Model Deter-mination of Wages in Large-Scale UK Macro Models

99 A. Brendemoen and H. Vennemo (1993): The MarginalCost of Funds in the Presence of External Effects

100 K.-G. Lindquist (1993): Empirical Modelling ofNorwegian Exports: A Disaggregated Approach

101 A.S. Jore, T. Skjerpen and A. Rygh Swensen (1993):Testing for Purchasing Power Parity and Interest RateParities on Norwegian Data

102 R. Nesbakken and S. StrOm (1993): The Choice of SpaceHeating System and Energy Consumption in NorwegianHouseholds (Will be issued later)

103 A. Aaheim and K. Nyborg (1993): "Green NationalProduct": Good Intentions, Poor Device?

104 K.H. Alfsen, H. Birkelund and M. Aaserud (1993):Secondary benefits of the EC Carbon/ Energy Tax

105 J. Aasness and B. Holtsmark (1993): Consumer Demandin a General Equilibrium Model for EnvironmentalAnalysis

106 K.-G. Lindquist (1993): The Existence of Factor Sub-stitution in the Primary Aluminium Industry: A Multi-variate Error Correction Approach on Norwegian PanelData

107 S. Kvemdokk (1994): Depletion of Fossil Fuels and theImpacts of Global Warming

108 K.A. Magnussen (1994): Precautionary Saving and Old-Age Pensions

109 F. Johansen (1994): Investment and Financial Con-straints: An Empirical Analysis of Norwegian Firms

110 K.A. Brekke and P. Boring (1994): The Volatility of OilWealth under Uncertainty about Parameter Values

111 M.J. Simpson (1994): Foreign Control and NorwegianManufacturing Performance

112 Y. Willassen and T.J. Klette (1994): CorrelatedMeasurement Errors, Bound on Parameters, and a Modelof Producer Behavior

113 D. Wetterwald (1994): Car ownership and private caruse. A microeconometric analysis based on Norwegiandata

114 K.E. Rosendahl (1994): Does Improved EnvironmentalPolicy Enhance Economic Growth? Endogenous GrowthTheory Applied to Developing Countries

115 L. Andreassen, D. Fredriksen and 0. Ljones (1994): TheFuture Burden of Public Pension Benefits. AMicrosimulation Study

116 A. Brendemoen (1994): Car Ownership Decisions inNorwegian Households.

117 A. Langorgen (1994): A Macromodel of LocalGovernment Spending Behaviour in Norway

118 K.A. Brekke (1994): Utilitarism, Equivalence Scales andLogarithmic Utility

119 K.A. Brekke, H. Luris and K. Nyborg (1994): SufficientWelfare Indicators: Allowing Disagreement inEvaluations of Social Welfare

120 T.J. Klette (1994): R&D, Scope Economies and Com-pany Structure: A "Not-so-Fixed Effect" Model of PlantPerformance

121 Y. Willassen (1994): A Generalization of Hall's Speci-fication of the Consumption function

122 E. Holmoy, T. Hzgeland and 0. Olsen (1994): EffectiveRates of Assistance for Norwegian Industries

123 K. Motu.' (1994): On Equity and Public Pricing inDeveloping Countries

124 J. Aasness, E. Eide and T. Skjerpen (1994): Crimi-nometrics, Latent Variables, Panel Data, and DifferentTypes of Crime

125 E. Biorn and T.J. Klette (1994): Errors in Variables andPanel Data: The Labour Demand Response to PermanentChanges in Output

126 I. Svendsen (1994): Do Norwegian Finns FormExtrapolative Expectations?

127 T.J. Klette and Z. Griliches (1994): The Inconsistency ofCommon Scale Estimators when Output Prices areUnobserved and Endogenous

128 K.E. Rosendahl (1994): Carbon Taxes and the PetroleumWealth

129 S. Johansen and A. Rygh Swensen (1994): TestingRational Expectations in Vector Autoregressive Models

130 T.J. Klette (1994): Estimating Price-Cost Margins andScale Economies from a Panel of Microdata

131 L. A. Grunfeld (1994): Monetary Aspects of BusinessCycles in Norway: An Exploratory Study Based onHistorical Data

132 K.-G. Lindquist (1994): Testing for Market Power in theNorwegian Primary Aluminium Industry

133 T. J. Klette (1994): R&D, Spillovers and Performanceamong Heterogenous Firms. An Empirical Study UsingMicrodata

134 K.A. Brekke and H.A. Gravningsmyhr (1994): AdjustingNNP for instrumental or defensive expenditures. Ananalytical approach

135 T.O. Thoresen (1995): Distributional and BehaviouralEffects of Child Care Subsidies

136 T. J. Klette and A. Mathiassen (1995): Job Creation, JobDestruction and Plant Turnover in NorwegianManufacturing

137 K. Nyborg (1995): Project Evaluations and DecisionProcesses

138 L. Andreassen (1995): A Framework for EstimatingDisequilibrium Models with Many Markets

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139 L. Andreassen (1995): Aggregation when Markets donot Clear

140 T. Skjerpen (1995): Is there a Business Cycle Com-ponent in Norwegian Macroeconomic Quarterly TimeSeries?

141 J.K. Dagsvik (1995): Probabilistic Choice Models forUncertain Outcomes

142 M. Ronsen (1995): Maternal employment in Norway, Aparity-specific analysis of the return to full-time andpart-time work after birth

143 A. Bruvoll, S. Glomsrod and H. Vennemo (1995): TheEnvironmental Drag on Long- term Economic Perfor-mance: Evidence from Norway

144 T. Bye and T. A. Johnsen (1995): Prospects for a Com-mon, Deregulated Nordic Electricity Market

145 B. Bye (1995): A Dynamic Equilibrium Analysis of aCarbon Tax

146 T. 0. Thoresen (1995): The Distributional Impact of theNorwegian Tax Reform Measured by Disproportionality

147 E. Holmoy and T. Hmgeland (1995): Effective Rates ofAssistance for Norwegian Industries

148 J. Aasness, T. Bye and H.T. Mysen (1995): WelfareEffects of Emission Taxes in Norway

149 J. Aasness, E. Biom and Terje Skjerpen (1995):Distribution of Preferences and Measurement Errors in aDisaggregated Expenditure System

150 E. Bowitz, T. Faehn, L A. Griinfeld and K. Moum(1995): Transitory Adjustment Costs and Long TermWelfare Effects of an EU-membership — The NorwegianCase

151 I. Svendsen (1995): Dynamic Modelling of DomesticPrices with Time-varying Elasticities and RationalExpectations

152 I. Svendsen (1995): Forward- and Backward LookingModels for Norwegian Export Prices

153 A. Langorgen (1995): On the SimultaneousDetermination of Current Expenditure, Real Capital, FeeIncome, and Public Debt in Norwegian LocalGovernment

154 A. Katz and T. Bye(1995): Returns to Publicly OwnedTransport Infrastructure Investment. A Cost Function/Cost Share Approach for Norway, 1971-1991

155 K. 0. Aarbu (1995): Some Issues About the NorwegianCapital Income Imputation Model

156 P. Boug, K. A. Mork and T. Tjemsland (1995): FinancialDeregulation and Consumer Behavior: the NorwegianExperience

157 B. E. Naug and R. Nymoen (1995): Import PriceFormation and Pricing to Market: A Test on NorwegianData

158 R. Aaberge (1995): Choosing Measures of Inequality forEmpirical Applications.

159 T. J. Klette and S. E. Fore (1995): Innovation and JobCreation in a Small Open Economy: Evidence fromNorwegian Manufacturing Plants 1982-92

160 S. Holden, D. Kolsrud and B. Vikoren (1995): NoisySignals in Target Zone Regimes: Theory and MonteCarlo Experiments

161 T. Hmgeland (1996): Monopolistic Competition,Resource Allocation and the Effects of Industrial Policy

162 S. Grepperud (1996): Poverty, Land Degradation andClimatic Uncertainty

163 S. Grepperud (1996): Soil Conservation as anInvestment in Land

164 K. A. Brekke, V. Iversen and J. Aune (1996): SoilWealth in Tanzania

165 J. K. Dagsvik, D.G. Wetterwald and R. Aaberge (1996):Potential Demand for Alternative Fuel Vehicles

166 J.K. Dagsvik (1996): Consumer Demand withUnobservable Product Attributes. Part I: Theory

167 J.K. Dagsvik (1996): Consumer Demand withUnobservable Product Attributes. Part II: Inference

168 R. Aaberge, A. BjOrklund, M. Jantti, M. Palme, P. J.Pedersen, N. Smith and T. Wennemo (1996): IncomeInequality and Income Mobility in the ScandinavianCountries Compared to the United States

169 K. Nyborg (1996): Some Norwegian Politicians' Use ofCost-Benefit Analysis

170 E. Berg, S. Kverndokk and K. E. Rosendahl (1996):Market Power, International CO2 Taxation andPetroleum Wealth

171 Rolf Aaberge, Ugo Colombino and Steinar Stns m(1996): Welfare Effects of Proportional Taxation:Empirical Evidence from Italy, Norway and Sweden

172 J.K. Dagsvik (1996): Dynamic Choice, MultistateDuration Models and Stochastic Structure

173 J.K. Dagsvik (1996): Aggregation in Matching Markets

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Discussion Papers

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